Homology cylinders and sutured manifolds for homologically fibered knots
Both homology cylinders and sutured manifolds [\textit{D. Gabai}, J. Differ. Geom. 18, 445--503 (1983; Zbl 0533.57013)] are a particular kind of cobordism between surfaces. Informally, a homology cylinder between compact homeomorphic surfaces \(S_1\) and \(S_2\) is a 3-manifold \(M\) with \(\partial M = S_1 \cup S_2\) and with the inclusions of each of \(S_1\) and \(S_2\) into \(M\) inducing isomorphisms on integer homology groups. A sutured manifold \((M,\gamma)\), on the other hand, is a compact 3-manifold \(M\) with annuli (called sutures) \(\gamma \subset \partial M\) dividing \(\partial M\) into two surfaces \(S_1\) and \(S_2\) which are given opposite normal orientations. Each of \(S_1\) and \(S_2\) is adjacent to each component of \(\gamma\) and on each component \(A\) of \(\gamma\) the orientations of \(S_1\) and \(S_2\) induce parallel orientations on the boundary components of \(A\). Given a knot \(K \subset S^3\) and a Seifert surface \(F\) for \(K\), we can give the exterior \(S^3 - N(F)\) of \(F\) a sutured manifold structure by taking a neighborhood of \(K \subset \partial N(F)\) to be the suture. If this sutured manifold is the product of \(F\) with an interval, then \(K\) is a fibered knot. A knot is ``homologically fibered'' if the degree of its normalized Alexander polynomial \(\Delta_K(t)\) is \(2g\) where \(g\) is the Seifert genus of \(K\) and \(\Delta_K(0) = \pm 1\).NEWLINENEWLINEThe paper under review considers the relationship between homology cylinders and sutured manifolds in several different contexts.NEWLINENEWLINESections 1, 2, and 3 are introductory. Section 4 applies invariants of homology cylinders to give criteria for a homologically fibered knot to be non-fibered. The two main invariants are the ``Magnus matrix'' [\textit{T. Sakasai}, Algebr. Geom. Topol. 8, No. 2, 803--848 (2008; Zbl 1156.57001)] and the ``\(\Gamma\)-torsion' (an invariant arising from algebraic K-theory). Section 5 gives a relationship between these invariants and the Alexander polynomial.NEWLINENEWLINESections 6 and 7 show how the Nakanishi index (a concept related to the \(\Gamma\)-torsion) can be used to give a lower bound on the handle numbers for the sutured manifolds obtained from genus 1 Seifert surfaces for doubled knots. The ``handle number'' of a sutured manifold is a measure of how far the sutured manifold is from being being a product. It is analogous to the notion of Heegaard genus for 3-manifolds with boundary. The authors show (in Corollary 1) that there exist homologically fibered knots having Seifert surfaces of genus 1 with arbitrarily large handle number. They follow this with an example (the pretzel knot \(9_{46}\)) showing that the lower bound on handle number arising from the Nakanishi index is sharp.NEWLINENEWLINEThe paper is very clear and concise, but with ample motivation and references to background material provided. Many examples are used to illustrate general principles.
scientific article; zbMATH DE number 6211477
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